Graph parameters and invariants of the orthogonal group

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Regts, G.

Publication date

2013

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Citation for published version (APA):

Regts, G. (2013). Graph parameters and invariants of the orthogonal group.

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Chapter 7

Edge-reflection positive

partition functions of

vertex-coloring models

Recall from Section 5.1 that the partition function of a vertex-coloring model is also the partition function of an edge-coloring model. In this chapter we char-acterize, using some fundamental results from geometric invariant theory, for which vertex-coloring models their partition functions are edge-reflection posi-tive, i.e., for which vertex-coloring models their partition functions are partition functions of real edge-coloring models.

This chapter is based on [54] except for Section 7.2, which is based on joint work with Jan Draisma [20, Section 6].

7.1 Introduction

In his paper [66] (see also [67]) on the characterization of partition functions of real edge-coloring models, Szegedy gave an explicit way to construct from a vertex-coloring model(a, B)overC an edge-coloring model h over C such that pa,B(H) =ph(H)for every H∈ G. We will now describe this construction.

Let (a, B) be an n-color vertex-coloring model over C. As B is symmetric we can write B = UTU for some n×k (complex) matrix U, for some k. Let u1, . . . , un ∈ Ck be the columns of U. Define the edge-coloring model h by

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i=1 u_ψ(v)(i)u_ψ(w)(i) =pa,B(G).

Where the last line follows from the fact that UTU = B. This completes the proof.

Note that the proof of Lemma 7.1 also shows that if an edge-coloring model h is of the form h=∑n_i=1aievu_i for certain nonzero ai ∈Ckand certain vectors

ui ∈ C, then the partition function of h is equal to the partition function of

(a, B)(onG), where a= (a1, . . . , an)and Bi,j =u_iTuj. We will sometimes abuse

notation and call h a vertex-coloring model.

Let(a, B)be a vertex-coloring model. If B is positive semidefinite, then h can be taken to be real valued, that is, in view of Theorem 5.2, pa,Bis edge-reflection

positive. Szegedy [66] moreover observed that for B=

a b b a

, with a, b≥0, pa,B is also edge-reflection positive. Clearly, for a=0 and b=1, this matrix is

not positive semidefinite. This made him ask the question, which partition of vertex-coloring models are edge-reflection positive (cf. [66, Question 3.2]).

In this chapter we give a complete characterization of edge-reflection posi-tive partition functions of vertex-coloring models overC. Let h = _∑n_i=1aievui for nonzero aiand distinct vectors ui ∈Ck. We start by giving a simple

charac-terization in terms of the ui and aifor phto be edge-reflection positive.

Lemma 7.2. Let u1, . . . , un ∈ Ck be distinct vectors, let a ∈ (C∗)n and let h :=

∑n

i=1aievu_i. Then h is an edge-coloring model overR if and only if the set{

ui

ai

|

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7.1. Introduction

Proof. Suppose first that the set{

ui

ai

|i=1, . . . , n}is closed under complex conjugation. Then for p ∈ R(R), h(p) = _∑n_i=1aip(ui) = ∑ni=1aip(ui) = h(p).

Hence, h(p) ∈R. So h is real valued.

Now the ’only if’ part. By possibly adding some vectors to{u1, . . . , un}and

extending the vector a with zero’s, we may assume that {u1, . . . , un}is closed

under complex conjugation. We must show that ui = uj implies ai = aj. We

may assume that u1 = u2. Using interpolating polynomials (cf. [17, Lemma

2.9]) we find p ∈ R(C) such that p(uj) = 1 if j = 1, 2 and 0 otherwise. Let

p0 := 1/2(p+p). Then p0 ∈ R(R) and consequently, h(p0) = ∑n_i=1aip(ui) =

a1+a2 ∈ R. Similarly, there exists q∈ R(C)such that q(u1) = i, q(u2) = −i

and q(uj) = 0 if j > 2. Setting q0 := 1/2(q+q)and applying h to it, we find

that i(a1−a2) ∈R. So we conclude that a1= a2. Continuing this way proves

the lemma.

Lemma 7.2 clearly explains why for B =

a b b a

, with a, b ≥0, we have that p_1,Bis edge-reflection positive. Here is another example.

Example 7.1. Let B=     0 2 0 2 2 0 2 0 0 2 0 4 2 0 4 0     and let U=     1 1 1 1 i −i i −i 0 0 1 1 0 0 i −i     . (7.2)

Then UTU= B, and so by Lemma 7.2, p(1,B)is equal to the partition function

of a real edge-coloring model.

One might think that Lemma 7.2 already gives the answer to Szegedy’s question, but the only thing it says is that if h := _∑n_i=1aievui for certain a ∈

(C∗)n and distinct u1, . . . , un ∈ Ck, then it is easy to check whether h is real.

In case h is not real valued, it does not rule out the possibility that there is another real-valued edge-coloring model h0 (with possible more than k colors) such that ph(H) =ph0(H)for all graphs H. Yet, surprisingly, a certain converse

to Lemma 7.2 holds. We need however a few more definitions to state it. For a k×n matrix U we denote its columns by u1, . . . , un. By U∗we denote

the conjugate transpose of U. Let, for any k,(·,·)denote the standard bilinear form on Ck. We call the matrix U nondegenerate if the span of u1, . . . , un is

nondegenerate with respect to(·,·). In other words, if rk(UTU) =rk(U). We think of vectors inCkas vectors inCl for any l≥k. We can now state the main result of this chapter. The proof will be given in Section 7.3.

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Theorem 7.3. Let (a, B) be a twin-free n-color vertex-coloring model. Let U be a nondegenerate k×n matrix such that UTU=B. Then the following are equivalent:

(i) pa,B =pyfor some real edge-coloring model y,

(ii) there exist l ≥k and g∈Ol(C)such that the set{

gui

ai

|i =1, . . . , n}is closed under complex conjugation,

(iii) there exist l ≥k and g∈Ol(C)such that∑ni=1aievgui is real.

If moreover, UU∗∈Rk×k_{, then we can take g equal to the identity in (ii) and (iii).}

Observe that if the set of columns of gU is closed under complex conjugation, then gU(gU)∗is real. So the existence of a nondegenerate matrix U such that UTU =B and UU∗is real, is a necessary condition for pa,B to be the partition

function of an edge-coloring model overR.

In case B is real, there is an easy way to obtain a k×n rank k matrix U, where k = rk(B), such that UU∗ ∈ Rk×k _{and U}T_U ₌ _{B, using the spectral}

decomposition of B. So by Theorem 7.3, we get the following characterization of partition functions of real vertex-colorings that are partition functions of real edge-coloring models. We will state it as a corollary.

Corollary 7.4. Let(a, B)be a twin-free n-color vertex-coloring model over R. Then

pa,B= phfor some real edge-coloring model h if and only if for each i∈ [n]there exists

j∈ [n]such that

(i) ai=aj,

(ii) for each eigenvector v of B with eigenvalue λ:

λ>0 ⇒ vi =vj,

λ<0 ⇒ vi = −vj.

We will now give some examples to illustrate Theorem 7.3 and Corollary 7.4.

Example 7.2. Let G be the graph on two nodes x1 and x2 with node weights

equal to 1; the loop at x1has weight 1; the loop at x2has weight 0 and the edge

x1x2has weight 1. Then hom(H, G)is equal to the number of independent sets

of H. Using Theorem 7.3, it is easy to see that the partition function of any real edge-coloring model can not be equal to hom(·, G). This can also be easily seen using Theorem 5.2.

Example 7.3. For any n ∈ N with n ≥2 consider Kn, the complete graph on

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7.2. Orbits of vertex-coloring models

H. The corresponding vertex-coloring model is(1, J−I), where1 denotes the all-ones vector, J the all-ones matrix and I the identity matrix. The eigenvalue

−1 of J−I has multiplicity n−1. Using that the eigenspace corresponding to

−1 is equal to1⊥, it is easy to see, using Corollary 7.4, that hom(·, Kn)is equal

to the partition function of a real edge-coloring model if and only if n=2. We do not know whether it is easy to deduce this from Theorem 5.2.

In view of Theorem 5.2, Example 7.3 shows that for each n ≥ 3 there ex-ists k, t ∈ N, k-fragments F1, . . . , Ft and λ ∈ Rt such that ∑i,j=1t λiλjhom(Fi∗

Fj, Kn) <0. It would be interesting to characterize for which (twin-free) graphs

G the invariant hom(·, G)is edge-reflection positive. By Corollary 7.4, this de-pends on spectral properties of G.

The remainder of this chapter is devoted to proving Theorem 7.3. The proof is based on a well-known generalization of the Hilbert-Mumford criterion, a fundamental result in geometric invariant theory. In the next section we use this criterion to characterize when the Ok(C)-orbit of a vertex-coloring model

is closed. In Section 7.3 we then use this result combined with some ideas of Kempf and Ness to give a proof of Theorem 7.3.

7.2 Orbits of vertex-coloring models

In this section we will work with a general algebraically closed fieldF of char-acteristic zero. Let k ∈ N and let V be a k-dimensional vectorspace over F

equipped with a nondegenerate symmetric bilinear form(·,·). Identify V with

Fk _{through the bilinear form. Let u}

1, . . . , un ∈V be distinct, and let a∈ (F∗)n.

Define the edge-coloring model h by h := _∑n_i=1aievui. In this section we will consider the Ok-orbit Okh≤e ⊂ FN

k

≤e _{for e} _∈ N (recall that h_≤e _{denotes the} restriction of h to the space of polynomials of degree at most e); we will char-acterize in terms of the ui when this orbit is closed. Our main tool will be a

well-known generalization of the Hilbert-Mumford criterion.

7.2.1 The one-parameter subgroup criterion

There is a beautiful criterion for closedness of orbits involving one-parameter subgroups of Ok, i.e., homomorphisms λ : F∗ → Ok of algebraic groups. We

call a basis v1, . . . , vk of V such that (vi, vj) = δk+1,i+j for all i, j, (i.e. so that

the Gram matrix of the basis has zeroes everywhere except ones on the longest anti-diagonal) a canonical basis. Let λ :F∗→Ok be a one-parameter subgroup.

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each t∈F∗, for some integral weights d1≥ · · · ≥dksatisfying di= −dk+1−i for

all i. This follows, for instance, from [25, Section 2.1.2] or [4, §23.4] (ignoring the subtle rationality issues there asF is algebraically closed) and the fact that all maximal tori are conjugate [4, §11.3]. Conversely, given a canonical basis v1, . . . , vkand such a sequence of di’s, the homomorphism λ :F∗→Okdefined

by λ(t)vi=tdiviis a one-parameter subgroup of Ok.

The one-parameter subgroup criterion says the following: let W be a finite-dimensional Ok-module, and let w ∈ W. Consider the orbit Okw ⊆ W. By

Theorem 4.7, the Zariski closure of this orbit contains a unique closed orbit C. Then there exists a one-parameter subgroup λ such that limt→0λ(t)w

ex-ists and is contained in C (the Hilbert-Mumford criterion considers the special case where C= {0}). Here the existence of the limit by definition means that the morphismF∗ →W, t 7→ λ(t)w extends toF. It then does so in a unique

manner, and the value at 0 is declared the limit. Put differently, just like V, the

λ-module W decomposes into a direct sum of weight spaces (cf. [25, Lemma

1.6.4]), and the condition is that all components of w in λ-weight spaces cor-responding to negative weights are zero, and the component of w in the zero weight space is the limit. We record the one-parameter subgroup criterion as a theorem.

Theorem 7.5. Let W be a finite dimensional Ok-module, let w ∈ W and let C be

the unique closed orbit contained in Okw. Then there exists a one-parameter subgroup

λ:F∗→Oksuch that the limit limt→0λ(t)w exists and is contained in C.

For a proof of Theorem 7.5 see e.g. [3, Theorem 4.2] or [32, Theorem 1.4].

Example 7.4. Recall the edge-coloring model h from Example 6.1 in Section 6.1,

h ∈k[x1, x2]∗ is zero on all polynomials of degree at least 2. The restriction of

h to the space of polynomials of degree at most 1 is an element of (V∗)∗ =

V, namely, equal to v1 := e1+ie2. This is an isotropic vector relative to the

bilinear form, and so is its complex conjugate v2 := e1−ie2. So the sequence

1/√2v1, 1/ √

2v2 forms a canonical basis of V. The linear map V →V scaling

v1with t∈F and v2with t−1is an element of the orthogonal group. Explicitly,

this gives the one-parameter subgroup

λ(t) = 1 2t 1+t2 i−it2 −i+it2 ₁₊_t2 ∈O2 (7.3)

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7.2. Orbits of vertex-coloring models

7.2.2 Application to vertex-coloring models

Here we will use the one-parameter subgroup criterion to characterize when the Ok-orbit of h≤eis closed.

We will need the following well-known result.

Proposition 7.6. Let u1, . . . , un ∈ V be nonzero. If w1, . . . , wn ∈ V are nonzero

vectors such that

(ui, uj) = (wi, wj)for all i, j=1, . . . , n, (7.4)

then there exists g∈Ok such that gui=vi for all i∈ [n].

For completeness we will sketch the proof.

Proof. Let U denote the span of the uiand W the span of the wi. If U =V, we

can just define a linear map g : V →V by ui 7→wi for each i. It is easy to see

that g is well defined and that g preserves the bilinear form, that is g∈Ok.

Next, if the bilinear form restricted to U is nondegenerate, then we can re-duce to the previous case by adding an ortonormal basis for U⊥to{u1, . . . , un}

and an orthonormal basis for W⊥to{w1, . . . , wn}.

Finally, if U is degenerate we can find i ∈ [n] such that(ui, uj) = 0 for all

j ∈ [n]. Let U0 ⊂ U and W0 ⊂ W be complements to ui and wi respectively.

Then we may choose u∈U0⊥such that(ui, u) =1 and such that(u, u) =0 (cf.

[36, XV, §9]). Similarly, we may choose w∈W0⊥such that(wi, w) =1 and such

that (w, w) = 0. Now add u to {u1, . . . , un} and w to {w1, . . . , wn} and note

that the dimension of U (and of W) increases by one. Now just proceed until U becomes nondegenerate so that we can reduce to the previous case.

Theorem 7.7. Let F = F, let u1, . . . , un ∈ V be distinct and let a ∈ (F∗)n. Let

h := _∑n_i=1aievui and let e ≥ 3n. Then the orbit Okh≤e is closed if and only if the restriction of the bilinear form to the span of the uiis nondegenerate.

Proof. Let U ⊂ V denote the space spanned by the ui. Suppose first that the

bilinear form restricted to U is degenerate. Then we may assume that(u1, ui)

is 0 for all i ∈ [n]. Define h0 = _∑n_i=2aievui. By Proposition 7.6, there exists for each ε> 0 , g∈ Ok such that gu1 = εu1 and gui = ui for i ≥2. This implies

that h0_≤e is contained in the closure of the orbit of h≤e. We will now show that

h0_≤eis not contained in the orbit h≤e.

Let I(h) ⊂ R be the set of polynomials p of degree at most n such that h(pq) =0 for all polynomials q of degree at most n−1. Then

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The inclusion ’⊇’ is clear. To see the other inclusion, let p1, . . . , pnbe

interpolat-ing polynomials at the ui, i.e., the pi are polynomials of degree n−1 such that

pi(uj) =δi,jfor all i, j=1, . . . , n (cf. [17, Lemma 2.9]). Then for a polynomial p

of degree at most n we have that deg(ppi) ≤2n−1≤e and h(ppi) =0 if and

only if p(ui) =0. This shows (7.5), which in turn implies

{u∈V| p(u) =0 for all p∈ I(h)} = {u1, . . . , un}. (7.6)

But since the uiare distinct, (7.6) applied to h0 implies that gh≤e6=h0≤e for any

g∈Ok, showing that the orbit of h≤eis not closed.

For the converse, assume that the bilinear form restricted to U is nonde-generate. We will prove that the Ok-orbit of h≤e is closed. Let λ : F∗ → Ok

be a one-parameter subgroup such that limt→0λ(t)h≤e exists. We will show

that it lies in the orbit of h≤e. Let v1, . . . , vk be a canonical basis of V with

λ(t)vj =tdjvj for weights d1≥ · · · ≥ dk. Let x1, . . . , xk be the basis of V∗ dual

to v1, . . . , vk. For any monomial xα, α∈Nk, we have (λ(t)h)(xα) =h(λ(t)−1xα) =h(tα1d1+...+αkdkxα) =tα·d

n

∑

i=1

aixα(ui), (7.7)

where α·d :=α1d1+. . .+αkdk. By assumption, if xαis a monomial of degree

at most e, the limit for t→0 in (7.7) exists. Note that this implies for|α| ≤e: α·d<0⇒h(xα) =

n

∑

i=1

aixα(ui) =0. (7.8)

In what follows, we exclude the trivial cases where k=0 and where k=1 and u1is the zero vector; in both of these cases the orbit of h is just a single point.

Next let b∈ {1, . . . , k}be the maximal index with xb(U) 6= {0}, and order

the uisuch that xb(u1), . . . , xb(ul) 6=0(l>0)and xb(ul+1), . . . , xb(un) =0. By

maximality of b, U is contained in the span of v1, . . . , vb. So if dbis nonnegative,

then limt→0λ(t)(u1, . . . , un) exists, and is by Proposition 7.6 contained in the

orbit of (u1, . . . , un). (Since U is nondegenerate, the equations describing the

orbit are given by (7.4).) Then also h≤eand limt→0λ(t)h≤elie in the same orbit.

So we may assume that db<0. (In particular, b>k/2.)

Since the coordinates xb+1, . . . , xk vanish identically on U, it follows that

U is contained in the subspace of V perpendicular to v1, . . . , vk−b. As U is

nondegenerate, it does not contain a nonzero linear combination of v1, . . . , vk−b.

This means, in particular, that the coordinates xk−b+1, . . . , xb together separate

the points u1, . . . , ul. Then so do the monomials xk−b+1x2b, . . . , xb−1x2b, x3b. Note

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7.3. Proof of Theorem 7.3

equals dk−b+1+2db =db <0 and from there the dot product decreases weakly

to the right). It follows that there exists a linear combination p of those cubic monomials for which p(u1), . . . , p(ul)are distinct and nonzero. Then, by (7.8)

and the fact that p(ul+1) = · · · = p(un) = 0, the vector (a1, . . . , al)T is in the

kernel of the Vandermonde matrix      p(u1) . . . p(ul) p(u1)2 . . . p(ul)2 .. . ... p(u1)l . . . p(ul)l      , (7.9)

since the degree of plis 3l≤e. This implies that a1, . . . , alare all zero, contrary

to the assumption that all ai are nonzero. This proves that the orbit of h≤e is

closed for e≥3n.

7.3 Proof of Theorem 7.3

In this section we give a proof of Theorem 7.3 using Theorem 7.7. We first need some preparations.

Let W ∈ Cl×n _{be any matrix and consider the function f}

W : Ol(C) → R

defined by

g7→tr(W∗g∗gW) =tr((gW)∗gW), (7.10) for g ∈Ol(C), where tr(M) denotes the trace of a matrix M and M∗the

con-jugate transpose of M. This function was introduced by Kempf and Ness [33] in the context of connected reductive linear algebraic groups acting on finite dimensional vector spaces. Note that fW is left-invariant under Ol(R) and

right-invariant under Stab(W) := {g ∈ Ol(C) | gW = W}. Let e ∈ Ol(C)

denote the identity. We are interested in the situation that the infimum of fW

over Ol(C)is equal to fW(e).

Lemma 7.8. The function fWhas the following properties:

(i) infg∈Ol(C) fW(g) = fW(e)if and only if WW

∗_∈_Rl×l_,

(ii) If WW∗∈Rl×l_{, then f}

W(e) = fW(g)if and only if g∈Ol(R) ·Stab(W).

Proof. We start by showing that

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By definition, a critical point of fWis a point g∈Ol(C)such that(D fW)g(X) =

0 for all X ∈ Tg(Ol(C)), where Tg(Ol(C)) is the tangent space of Ol(C)at g

and where(D fW)gis the derivative of fWat g. It is well known that the tangent

space of Ol(C)at e is the space of skew-symmetric matrices, i.e., Te(Ol(C)) = {X∈Cl×l _|_XT₊_X₌₀_}_{. It is easy to see that the derivative of f}

Wat e is the

R-linear map(D fW)e ∈HomR(Cl×l,R)defined by Z7→tr(W∗(Z+Z∗)W). Now

let Z be skew-symmetric and write Z = X+iY, with X, Y ∈ Rl×l_{. Note that}

Z is skew-symmetric if and only if both X and Y are skew-symmetric. Write W=V+iT with V, T∈Rl×l_{. Then}₍_{D f}

W)e(Z)is equal to

tr((VT−iTT)(X+iY+XT−iYT)(V+iT)) = 2tr((VT−iTT)iY(V+iT))

= 2tr(TTYV) −2tr(VTYT) =4tr(TTYV), (7.12) where we use that X and Y are skew symmetric, and standard properties of the trace. So D fe(Z) =0 for all skew symmetric Z if and only if VTT =TVT. That

is, if and only if WW∗∈Rl×l_{. This shows (7.11).}

By a result of Kempf and Ness (cf. [33, Theorem 0.1]) we can now conclude that (i) and (ii) hold. However, we will give an independent and elementary proof.

First the proof of (i). Note that (7.11) immediately implies that fWdoes not

attain a minimum at e if WW∗∈/Rl×l_{. (This follows easily from the method of}

Lagrange multipliers.) Conversely, suppose WW∗ ∈ Rl×l_{. Since WW}∗ _{is real}

and positive semidefinite there exists a real matrix V such that WW∗ =VVT. Now note that, by the cyclic property of the trace, fW(g) =tr(g∗gWW∗). So we

have fW= fV. Let I denote the identity matrix. Take any g=X+iY∈Ol(C),

where X, Y ∈ Rl×l_{. Using that X}T_X₋_YT_Y ₌ _{I, and the fact that f}

W is real

valued, we find that

fW(g) = tr((XTX+YTY)VVT) =tr(VVT) +2tr(YTYVVT)

= 2tr(VVT) +tr(YV(YV)T) ≥tr(VVT) = fW(e). (7.13)

This proves (i).

Next, suppose that fW(g) = fW(e)for some g ∈Ol(C). Again, since WW∗

is real and positive semidefinite there exists a real matrix V such that WW∗=

VVT. Moreover, the span of the columns of V is equal to the span of the columns of W. This implies that Stab(V) =Stab(W). Now write g = X+iY, with X, Y∈Rl×l_{. As, by (7.13), f}

W(g) = fW(e)if and only if YV =0, it follows

that gV = XV+iYV = XV is a real matrix. Let v1, . . . , vn be the columns of

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all i, j, and since the gvi and the vi are real, there exists g1 ∈ Ol(R)such that

g1gV = V. This implies that g ∈ Ol(R) ·Stab(V). This finishes the proof of

(ii).

For any e, leth·,·idenote the Hermitian inner product onCNl≤e _{induced by} the standard Hermitian inner product onLe

i=1(Cl)⊗i, by viewing elements of

CNl≤e _{as symmetric tensors. The next proposition has as conclusion a special} case of Theorem 0.2 in [33].

Proposition 7.9. Let h be any l-color edge-coloring model. Let Cebe the unique closed

orbit in Ol(C)h≤e. Then there exists h0_≤e∈Cesuch that

inf

g∈Ol(C)

hgh≤e, gh≤ei ≥ hh0≤e, h0≤ei. (7.14)

Moreover, the infimum is attained if and only if h≤e ∈Ce.

Proof. Clearly, the infimum is attained at some g ∈ Ol(C)if h≤e ∈ Ce. So we

can take h0 =gh.

Now assume that h≤e∈/Ce. Fix any g∈Ol(C), write y :=gh and, as in the

proof of Theorem 7.7, let λ :C∗ →Ol(C)be a one-parameter subgroup such

that limt→0λ(t)y≤e = y≤e0 ∈ Ce. Let v1, . . . , vl be a canonical basis ofCl with

λ(t)vj = tdjvj for weights d1 ≥ · · · ≥ dl. Let x1, . . . , xl be the basis of (Cl)∗

dual to v1, . . . , vl. Recall from (7.7) that for any monomial xα, α∈ Nl, we have (λ(t)y)(xα) =tα·dy(xα). Since, by assumption, the limit limt→0tα·dy(xα)exists

for|α| ≤e, this implies:

y0_≤e(xα_{) =}    0(=y(xα₎₎ _{if α}_·_d_<_0, y(xα₎ _{if α}_·_d₌_0, 0 if α·d>0. (7.15)

For e0 ≤ e and φ : [e0] → [l] let φ·d := α·d, for α ∈ Nl such that xφ =

x_φ(₁₎· · ·x_φ(l) =xα_{. Then, as y} ≤e6=y0≤e, by (7.15), hy≤e, y≤ei =

∑

e0=0,...,e φ:[e0]→[l] φ·d≥0 y(xφ₎_y₍_xφ_{) >}

∑

e0=0,...,e φ:[e0]→[l] φ·d=0 y(xφ₎_y₍_xφ_{) = h}_y0 ≤e, y0≤ei. (7.16)

So for each g∈Ol(C)we can find y0≤e∈Cesuch thathgh≤e, gh≤ei > hy0≤e, y0≤ei,

proving the first statement. This moreover implies that the infimum is not attained if h≤e∈/Ce, finishing the proof.

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We need one more lemma before we can prove Theorem 7.3.

Lemma 7.10. Let h:=_∑n_i=1aievui ∈ R(C)

∗_{, with a}_{∈ (}_C∗₎n_{and u}

1, . . . , un ∈Ck

distinct. Suppose the bilinear form restricted to the span of the ui is nondegenerate. If

y is a real l-color edge-coloring model such that ph(H) = py(H)for all H ∈ G, then

there exists g∈Ol(C)such that gh=y.

Proof. We may assume that l ≥ k. Recall that in case l > k we add colors to h. This is done by appending the ui’s with zero’s. Note that the bilinear form

restricted to the span of the ui remains nondegenerate. Then, by Theorem 7.7,

for each d≥3n, the orbit Olh≤dis equal to the unique closed orbit Cd. We will

now show that the orbit of y≤d is also equal to Cdfor any d.

For any e ≤ d, Ol(C) embeds naturally into Ole(C). Let g ∈ O_le(C), and write g=X+iY, with X, Y∈Rle×le

. Then, using that XTX−YT_Y₌_I, hgye, gyei = hXye, Xyei + hYye, Yyei (7.17)

= hye, yei +2hYye, Yyei ≥ hye, yei.

As this holds for any e≤ d, we can now conclude by Proposition 7.9 that the orbit of y≤dis closed.

We now claim that this implies that there exists g∈Ol(C)such that gh=y.

Indeed, since Stab(y≤d) = ∩d0_≤dStab(y_≤d0)and since O_l(C)is Noetherian, there

exists d1 ≥ 3n such that Stab(y≤d1) = ∩d∈NStab(y≤d). Recall that we have a canonical bijection from Ol(C)/Stab(y≤d)to Cdgiven by

gStab(y≤d) 7→gy≤d (7.18)

(cf. the proof of Theorem 6.11). This implies that for any d≥d1, if g∈Ol(C)

is such that gy≤d =h≤d, then also gy=h. This proves the lemma.

Now we can give a proof of Theorem 7.3.

Theorem 7.3. Let (a, B) be a twin-free n-color vertex-coloring model. Let U be a

nondegenerate k×n matrix such that UT_U₌_{B. Then the following are equivalent:}

(i) pa,B =pyfor some real edge-coloring model y,

(ii) there exist l ≥k and g∈Ol(C)such that the set{

gui

ai

|i =1, . . . , n}is closed under complex conjugation,

(iii) there exist l ≥k and g∈Ol(C)such that∑ni=1aievgui is real.

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Proof. Observe that since (a, B) is twin free, the columns of U are distinct. Lemma 7.2 implies the equivalence of (ii) and (iii) for the same g and l in (ii) and (iii). Moreover, since(gU)TgU=UTgTgU =UTU=B, for any g∈Ol(C),

Lemma 7.1 shows that (iii) implies (i).

Let u1, . . . , un be the columns of U and let h := ∑ni=1aievui. We will now prove that (i) implies (iii). Let y be a real l-color edge-coloring model such that pa,B = py. Since U is nondegenerate, we may assume, by Lemma 7.10, that

y= gh for some g∈Ol(C). Now note that gh =∑ni=1aievgui. This shows that (i) implies (iii).

Now assume that UU∗ ∈ Rk×k_{. We will show that (i) implies (iii) with}

g = e. Let y be a real l-color edge-coloring model such that pa,B = py. Just as

above, we may assume that y =∑n_i=1aievgu_i, for some g ∈ Ol(C). Lemma 7.2

implies that the set{gui} is closed under complex conjugation. This implies

that gU(gU)∗ ∈ Rl×l_{. So by Lemma 7.8 (i) the infimum of f}

gU is attained at

e. Equivalently, the infimum of fU is attained at g. Since UU∗ ∈ Rk×k, this

implies, by Lemma 7.8 (ii), that g∈Ol(R) ·Stab(U). Hence g=g1·s for some

g1∈Ol(R)and s∈Stab(U). Now note that since sh=h we have that h=g−11 y